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porcupine6789
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Can someone explain these to me?
Feldoh said:If w is in the same direction as v it must be some multiple of v. In other words [tex]w = \lambda v[/tex] where lambda is just some scalar value.
If this is the case we can say that [tex]Av = \lambda v[/tex] where lambda is the eigenvalue and v is an eigenvector of the matrix A
maxverywell said:I have some questions about this. If w is in the same direction as v this means that A (which is an operator represented by matrix A) don't change the direction of w, i.e the A don't rotate the vector w. How do we call such operators (or matrices)? Are they only linear transformations?
HallsofIvy said:A linear operator on a vector space is a function, F, such that F(au+ bv)= aF(u)+ bF(v) where u and v are vectors and a and b are scalars.
A matrix is, of course, an array of numbers with specific addition and multiplication defined.
We can think of n by m matrices as linear transformations from the specific vector space [itex]R^n[/itex] to the vector space [itex]R^m[/itex].
More generally, if F is a linear transformation from a vector space of dimension n to a vector space of dimension m, then, given a specific ordered basis, it can be represented as an n by m matrix.
But linear transformations are not the same as matrices. For one thing, the correspondence depends on a specific basis (the matrices representing the same linear transformation in different bases are "similar" matrices however). Further, there exist infinite dimensional vector spaces in which linear transformations cannot be represented by matrices. One example is the vector space of all polynomials (with the usual definition of addition of polynomials and multiplication by numbers) with the linear operator being differentiation of the polynomial.
Eigenvectors and eigenvalues are mathematical concepts that are commonly used in linear algebra. Eigenvectors are special vectors that do not change direction when multiplied by a linear transformation. Eigenvalues are the corresponding scalars that represent how much the eigenvectors are scaled by the transformation.
Eigenvectors and eigenvalues have a wide range of applications in various fields such as physics, engineering, and data analysis. They are used to solve systems of differential equations, find stable systems in physics, and reduce the dimensionality of data in machine learning.
To find the eigenvectors and eigenvalues of a matrix, you need to solve the characteristic equation, which is obtained by setting the determinant of the matrix minus a multiple of the identity matrix equal to 0. The resulting eigenvalues can then be used to find the corresponding eigenvectors.
The eigenvector with the largest eigenvalue is known as the dominant eigenvector and is of particular interest because it represents the direction of the greatest change in a system. This is useful in applications such as principal component analysis, where the dominant eigenvector represents the most important dimension of a dataset.
Yes, a matrix can have multiple sets of eigenvectors and eigenvalues, as long as they are linearly independent. This means that there can be multiple solutions to the characteristic equation, resulting in different eigenvalues and eigenvectors. However, each set of eigenvectors and eigenvalues will still satisfy the definition of an eigenvector and eigenvalue for that particular matrix.